Proof Complexity of Substructural Logics

In this paper, we investigate the proof complexity of a wide range of substructural systems. For any proof system \(\mathbf{P}\) at least as strong as Full Lambek calculus, \(\mathbf{FL}\), and polynomially simulated by the extended Frege system for some infinite branching super-intuitionistic logic, we present an exponential lower bound on the proof lengths. More precisely, we will provide a sequence of \(\mathbf{P}\)-provable formulas \(\{A_n\}_{n=1}^{\infty}\) such that the length of the shortest \(\mathbf{P}\)-proof for \(A_n\) is exponential in the length of \(A_n\). The lower bound also extends to the number of proof-lines (proof-lengths) in any Frege system (extended Frege system) for a logic between \(\mathsf{FL}\) and any infinite branching super-intuitionistic logic. We will also prove a similar result for the proof systems and logics extending Visser's basic propositional calculus \(\mathbf{BPC}\) and its logic \(\mathsf{BPC}\), respectively. Finally, in the classical substructural setting, we will establish an exponential lower bound on the number of proof-lines in any proof system polynomially simulated by the cut-free version of \(\mathbf{CFL_{ew}}\).